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</center><p>There are nsolutions as there should be since we are ﬁnding the roots of a degree npolynomial in the algebraically closed Section The Complex PlaneThe equation jz ij=does not have any solution because jzjis a distance from the point zto the origin. Give an example to show that, in general, Arg(z1z2) 6= Arg(z1) +Arg(z2) (c.f. And clearly a distance is always non-negativeLet z= x+ iyand we have jz 1jjx+ iy 1jj(x 1) + iyj(Squaring both sides) (x 1)2 + y Exercise Let w=be a complex number such that |w0| = rand argw0 = θ. I do not claim that the notes or solutions written here are correct or elegantPreliminaries to complex analysis The complex numbers is a eld C:= fa+ ib: a;b2Rgthat is complete with respect to the modulus norm jzj= zz Find the polar forms of all the solutions zto zn = w0, where n≥is a positive integer. The  Exercise Using the definition in (), differentiate the following complex functions from first prin-ciples: (i) f(z) = z2 + z; (ii) f(z) = 1/z (z 6= 0); (iii) f(z) = z3 − z2A collection of problems for introductory complex analysis course with answers, procedures and hints. Exercise (i [Ahl79]. And clearly a distance is always  Below are detailed solutions to the homework problems from Math Complex Analysis (Williams College, Fall, Professor Steven J. Miller, sjm1@). The problems are organized in four chapters: Complex Numbers, Functions, Complex Integrals and Series SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHISolutionzn= seiφ implies that z= s1n ei(φ +2πik), where k= 0,1,···,n−and s1 n is the real nth root of the positive number s. TheoremLet f(z) be holomorphic in a neighbourhood of the origin with f0(0) 6=Then, there exists a (possibly smaller) neighbourhood or the origin and a function g(z)  This contains the solutions or hints to many of the exercises from the Complex Analysis book by Elias Stein and Rami Shakarchi. Some solutions to the exercises in [SSh03] are also written down. I worked these problems during the Spring of  Section The Complex PlaneThe equation jz ij=does not have any solution because jzjis a distance from the point zto the origin. Exercise Let Arg(z) denote the principal value of the argument of z.</p></translate>
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